4.2 Pulsar timing

Contrary to the solar system case, the dependence of the gravitational binding energy cannot be neglected while computing the time variation of the period. Here two approaches can be followed; either one sticks to a model (e.g., scalar-tensor gravity) and compute all the effects in this model or one has a more phenomenological approach and tries to put some model-independent bounds.

Eardley [177Jump To The Next Citation Point] followed the first route and discussed the effects of a time variation of the gravitational constant on binary pulsar in the framework of the Brans–Dicke theory. In that case, both a dipole gravitational radiation and the variation of G induce a periodic variation in the pulse period. Nordtvedt [386Jump To The Next Citation Point] showed that the orbital period changes as

˙ [ ] ˙ P- = − 2 + 2(m1c1-+-m2c2-) +-3(m1c2-+-m2c1-) G- (137 ) P m1 + m2 G
where ci ≡ δ ln mi ∕δlnG. He concluded that for the pulsar PSR 1913+16 (m1 ≃ m2 and c1 ≃ c2) one gets
P˙ G˙ P = − [2 + 5c]G , (138 )
the coefficient c being model dependent. As another application, he estimated that −10 cEarth ∼ − 5 × 10, cMoon ∼ − 10− 8 and cSun ∼ − 4 × 10 −6 justifying the formula used in the solar system.

Damour et al. [127] used the timing data of the binary pulsar PSR 1913+16. They implemented the effect of the time variation of G by considering the effect on ˙ P∕P. They defined, in a phenomenological way, that G˙∕G = − 0.5δP˙∕P, where δP˙ is the part of the orbital period derivative that is not explained otherwise (by gravitational waves radiation damping). This theory-independent definition has to be contrasted with the theory-dependent result (138View Equation) by Nordtvedt [386]. They got

˙ −11 −1 G ∕G = (1.0 ± 2.3) × 10 yr . (139 )
Damour and Taylor [137] then reexamined the data of PSR 1913+16 and established the upper bound
G˙∕G < (1.10 ± 1.07) × 10− 11 yr−1. (140 )
Kaspi et al. [282] used data from PSR B1913+16 and PSR B1855+09 respectively to get
G˙∕G = (4 ± 5) × 10−12 yr−1 (141 )
and
G˙∕G = (− 9 ± 18) × 10 −12 yr −1, (142 )
the latter case being more “secure” since the orbiting companion is not a neutron star.

All the previous results concern binary pulsars but isolated ones can also be used. Heintzmann and Hillebrandt [248Jump To The Next Citation Point] related the spin-down of the pulsar JP1953 to a time variation of G. The spin-down is a combined effect of electromagnetic losses, emission of gravitational waves, possible spin-up due to matter accretion. Assuming that the angular momentum is conserved so that I∕P = constant, one deduces that

|| ( ) P˙| = -dln-I ˙G-. (143 ) P || d lnG G G
The observational spin-down can be decomposed as
˙|| ˙|| ˙|| ˙|| P-|| = P-|| + P-|| + P-|| . (144 ) P | P | P | P | obs mag GW G
Since P˙∕Pmag and P˙∕PGW are positive definite, it follows that P˙∕Pobs ≥ P˙∕PG so that a bound on G˙ can be inferred if the main pulse period is the period of rotation. Heintzmann and Hillebrandt [248] then modelled the pulsar by a polytropic (P ∝ ρn) white dwarf and deduced that d ln I∕d lnG = 2 − 3n∕2 so that ˙ − 10 −1 |G ∕G| < 10 yr. Mansfield [344] assumed a relativistic degenerate, zero temperature polytropic star and got that, when G˙ < 0, 0 ≤ − ˙G∕G < 6.8 × 10−11 yr−1 at a 2σ level. He also noted that a positive G˙ induces a spin-up counteracting the electromagnetic spin-down, which can provide another bound if an independent estimate of the pulsar magnetic field can be obtained. Goldman [233], following Eardley [177], used the scaling relations N ∝ G −3∕2 and M ∝ G −5∕2 to deduce that 2dln I∕dln G = − 5 + 3d ln I∕d lnN. He used the data from the pulsar PSR 0655+64 to deduce that the rate of decrease of G was smaller than
0 ≤ − G˙∕G < 5.5 × 10− 11 yr− 1. (145 )
The analysis [516] of 10 years high precision timing data on the millisecond pulsar PSR J0437-4715 has allowed to improve the constraint to
|G˙∕G | < 2.3 × 10−11 yr−1. (146 )

Recently, it was argued [266, 432] that a variation of G would induce a departure of the neutron star matter from β-equilibrium, due to the changing hydrostatic equilibrium. This would force non-equilibrium β-processes to occur, which release energy that is invested partly in neutrino emission and partly in heating the stellar interior. Eventually, the star arrives at a stationary state in which the temperature remains nearly constant, as the forcing through the change of G is balanced by the ongoing reactions. Comparing the surface temperature of the nearest millisecond pulsar, PSR J0437-4715, inferred from ultraviolet observations, two upper limits for variation were obtained, ˙ −10 −1 |G∕G | < 2 × 10 yr, direct Urca reactions operating in the neutron star core are allowed, and ˙ −12 − 1 |G ∕G | < 4 × 10 yr, considering only modified Urca reactions. This was extended in [302] in order to take into account the correlation between the surface temperatures and the radii of some old neutron stars to get |G˙∕G | < 2.1 × 10−11 yr−1.


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